it depends on the type of insect, a lot of them die, but lay eggs in the ground or in rotten logs, and those eggs make it through the winter, then hatch in the spring/summer.(grasshoppers and katydids do this)

(some overwinter as larvae or pupae, usually underground. (beetles are a prime example, Saturniidae, a group of moths do this, too, and they're not the only ones.)

and some migrate, like the Monarch butterfly.